Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Scheme.exists_isOpenCover_and_isAffine

∀ {I : Type u} [inst : CategoryTheory.Category.{u, u} I] (D : CategoryTheory.Functor I AlgebraicGeometry.Scheme)
  (c : CategoryTheory.Limits.Cone D) (hc : CategoryTheory.Limits.IsLimit c) [CategoryTheory.IsCofiltered I]
  [∀ {i j : I} (f : i ⟶ j), AlgebraicGeometry.IsAffineHom (D.map f)] [∀ (i : I), CompactSpace ↥(D.obj i)]
  [∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)] {J : Type u_1} (U : J → c.pt.Opens),
  TopologicalSpace.IsOpenCover U →
    (∀ (i : J), AlgebraicGeometry.IsAffineOpen (U i)) →
      ∃ i s V,
        TopologicalSpace.IsOpenCover V ∧
          ∀ (j : ↥s),
            AlgebraicGeometry.IsAffineOpen (V j) ∧ U ↑j = (TopologicalSpace.Opens.map (c.π.app i).base).obj (V j)

Suppose { Xᵢ } is an inverse system of qcqs schemes with affine transition maps. Then any affine open cover of lim Xᵢ comes from a finite level.

Defined in

Mathlib.AlgebraicGeometry.AffineTransitionLimit

Cited by

1 results in Mathlib

Foundations

Depth 244 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.IsCofilteredAlgebraicGeometry.IsAffineHomCompactSpaceQuasiSeparatedSpace